I think a necessary point to understand Bruce and Dick, the bond between them, and how being a father saved Bruce in many ways, is to understand that after the death of his parents, all the adult figures in Bruce's life failed him.
Bruce grew up a not only a orphan traumatized child, but also a child who didn't feel he was loved. He had a guardian, a tutor, and a billionaire inheritance, so obviously he was well taken care of and never lacked anything, but Alfred treated him like his young master who was his responsibility after the death of his former employers, and Leslie, despite being legally his adoptive mother, was absent for most of his childhood.
He didn't have the emotional support and did not grow up in the warm and familial environment he needed. He was a depressed child and became a depressed man.
Dick's adoption is an altruistic and empathetic act by Bruce; he knows how terrible, how lonely it is to spend your childhood feeling that the only people who loved you and cared deeply about you are dead.
Bruce gave Dick what Alfred and Leslie weren't able to give him.
Dick grew up a happy child, Dick had a good childhood and grew up knowing that Bruce was there for him even when they fought, Dick grew up knowing that Bruce stayed because he wanted to and cared about Dick. He is Bruce's greatest success because he's proof that it's possible to grieve without letting your whole life be swallowed up by your trauma.
Fathering Dick also helped Bruce; Dick made Bruce a happier man. When you take care of someone, especially someone young, that person tends to develop affection for you as well. And human beings are social beings, they like to feel that their relationships are reciprocal, and it is by loving Dick like a son that Bruce feels loved back for the first time since his parents died.
It's a similar situation with Tim. Sometimes, when we say that being a father saved Bruce, some people understand that this means some role dynamic was reversed, that one of those children had to take on some responsibility for Bruce at some point, but that's not what it means. Caring for someone can be healing, to love and to be loved back can be healing, sharing your daily life with someone who is dear to you and feeling that you are also important for them can be healing.
None of Bruce's children was ever parentificated for the sake of Bruce, saving them is what saved him.
Is not only Batman, the caped crusader, who is making for others the difference that would have mattered to him, but also Bruce, the father and the son, and the lonely child.
Batman: Dark Victory #9
Batman (1940) #438
Detective Comics (2016) #1074
Batman (1940) #439
